The types of derivatives and bifurcation in fractional mechanics

Fractional calculus appears to be a powerful tool of solid mechanics. In recent years several papers have been published including various forms of non-localities. Two basic fields can be distinguished, there are non-locality in space and in time. When term non-locality is used in its original meaning the value of some quantity in an internal point of the body is determined by the values of other quantities in a whole region around that location. The second one means to include hereditary effects like non-classical viscosity and study fractional visco-elasticity or visco-plasticity. In non-linear stability investigation the way of the loss of stability is studied and classified as a generic static or dynamic bifurcation. To do that step some kind of regular condition is necessary. This condition is connected with non-locality. Generally such behaviour is a result of viscous (time derivative) and gradient dependent terms in the constitutive equations. Such derivatives are not always of first order. There are materials where tests justify models with fractional order derivatives. Moreover, there are (for example Riemann-Liouville) type of fractional derivatives which are non-local itself. Thus by defining strain by fractional derivation of the displacement field a non-local quantity appears instead of the conventional (local) strain. In such a way various versions of non-localities are obtained by using various types of fractional derivatives. The paper studies how the selection of that fractional derivative effects the way of the loss of stability. Basically the study aims constitutive modelling via instability phenomena, that is, by observing the way of loss of stability of some material we can be informed about the form of fractional derivative in its mathematical model.